DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation61046710.1155/2010/610467610467Research ArticleBoundedness and Global Attractivity of a Higher-Order Nonlinear Difference EquationJiaXiu-Mei1, 2LiWan-Tong2ZhangGuang1Department of MathematicsHexi UniversityZhangye, Gansu 734000Chinahxu.edu.cn2School of Mathematics and StatisticsLanzhou UniversityLanzhou, Gansu 730000Chinalzu.edu.cn201014022010201005112009040220102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: yn+1=(r+pyn+ynk)/(qyn+ynk), n0, where the parameters p,q,r(0,),k{1,2,3,} and the initial conditions yk,,y0(0,). We show that the unique positive equilibrium of this equation is a global attractor under certain conditions.

1. Introduction and Preliminaries

Our aim in this paper is to study the dynamical behavior of the following rational difference equation

yn+1=r+pyn+yn-kqyn+yn-k,n0, where p,q,r(0,), 0{0,1,}, k{1,2,3,} and the initial conditions y-k,,y0(0,).

When k=1, (1.1) reduces to

yn+1=r+pyn+yn-1qyn+yn-1,n0.

In  (see also ), the authors investigated the global convergence of solutions to (1.2) and they obtained the following result.

Theorem 1.1.

Let p, q and r be positive numbers. Then every solution of (1.2) converges to the unique equilibrium or to a prime-two solution.

The main purpose of this paper is to further consider the global attractivity of all positive solutions of (1.1). That is to say, we will prove that the unique positive equilibrium of (1.1) is a global attractor under certain conditions (see Theorem 4.10).

For the general theory of difference equations, one can refer to the monographes  and . For other related results on nonlinear difference equations, see, for example, .

For the sake of convenience, we firstly present some definitions and known results which will be useful in the sequel.

Let I be some interval of real numbers and let f:I×II be a continuously differentiable function. Then for initial conditions x-k,,x0I, the difference equation

xn+1=f(xn,xn-k),n0 has a unique solution {xn}n=-k.

A point x̅ is called an equilibrium of (1.3) if

x̅=f(x̅,x̅).

That is, xn=x̅ for n0 is a solution of (1.3), or equivalently, x̅ is a fixed point of f.

An interval JI is called an invariant interval of (1.3) if

x-k,,x0JxnJn0.

That is, every solution of (1.3) with initial conditions in J remains in J.

Let

P=fu(x̅,x̅),Q=fv(x̅,x̅)

denote the partial derivatives of f(u,v) evaluated at an equilibrium x̅ of (1.3). Then the linearized equation associated with (1.3) about the equilibrium x̅ is

zn+1=Pzn+Qzn-k,n=0,1,, and its characteristic equation is

λk+1-Pλk-Q=0.

Lemma 1.2 (see [<xref ref-type="bibr" rid="B10">3</xref>]).

Assume that P,QR and k{1,2,}. Then |P|+|Q|<1 is a sufficient condition for asymptotic stability of the difference equation (1.7). Suppose in addition that one of the following two cases holds:

k odd and Q>0,

k even and PQ>0.

Then (1.9) is also a necessary condition for the asymptotic stability of the difference equation (1.7).

The following result will be useful in establishing the global attractivity character of the equilibrium of (1.1), and it is a reformulation of [2, 7].

Lemma 1.3.

Suppose that a continuous function f:[a,b]×[a,b][a,b] satisfies one of (i)–(iii):

f(x,y) is nonincreasing in x,y, and (m,M)[a,b]×[a,b],(f(m,m)=M,f(M,M)=m)m=M,

f(x,y) is nondecreasing in x and nonincreasing in y, and (m,M)[a,b]×[a,b],(f(m,M)=m,f(M,m)=M)m=M,

f(x,y) is nonincreasing in x and nondecreasing in y, and (m,M)[a,b]×[a,b],(f(M,m)=m,f(m,M)=M)m=M.

(Note that for k odd this is equivalent to (1.3) having no prime period-two solution)

Then (1.3) has a unique equilibrium in [a,b] and every solution with initial values in [a,b] converges to the equilibrium.

This work is organized as follows. In Section 2, the local stability and periodic character are discussed. In Section 3, the boundedness, invariant intervals of (1.1) are presented. Our main results are formulated and proved in Section 4, where the global attractivity of (1.1) is investigated.

2. Local Stability and Period-Two Solutions

The unique positive equilibrium of (1.1) is

y̅=(1+p)+(1+p)2+4r(1+q)2(1+q).

The linearized equation associated with (1.1) about y̅ is

zn+1-(p-q)y̅-qr(q+1)[r+(p+1)y̅]zn+(p-q)y̅+r(q+1)[r+(p+1)y̅]zn-k=0,

and its characteristic equation is

λk+1-(p-q)y̅-qr(q+1)[r+(p+1)y̅]λk+(p-q)y̅+r(q+1)[r+(p+1)y̅]=0.

From this and Lemma 1.2, we have the following result.

Theorem 2.1.

Assume that p,q,r(0,) and initial conditions y-k,,y0(0,). Then the following stataments are true.

If (p-q)y̅-qr0,(p-3q-pq-1)y̅<2qr, then the unique positive equilibrium y̅ of (1.1) is locally asymptotically stable;

If (p-q)y̅-qr<0<(p-q)y̅+r, then the unique positive equilibrium y̅ of (1.1) is locally asymptotically stable. In particular, if k is even, then the equilibrium y̅ is locally asymptotically stable if and only if (2.5) holds;

If (p-q)y̅+r0, then the unique positive equilibrium y̅ of (1.1) is locally asymptotically stable.

In the following, we will consider the period-two solutions of (1.1).

Let

,ϕ,ψ,ϕ,ψ,

be a period-two solution of (1.1), where ϕ and ψ are two arbitrary positive real numbers.

If k is even, then yn=yn-k, and ϕ and ψ satisfy the following system:

ϕ=r+pψ+ψqψ+ψ,ψ=r+pϕ+ϕqϕ+ϕ,

then (ϕ-ψ)(p+1)=0, we have ϕ=ψ, which is a contradiction.

If k is odd, then yn+1=yn-k, and ϕ and ψ satisfy the following system:

ϕ=r+pψ+ϕqψ+ϕ,ψ=r+pϕ+ψqϕ+ψ,

then ϕ+ψ=1-p, ϕψ=p(1-p)/(q-1). By calculating, (1.1) has prime period-two solution if and only if

p<1,q>1,4r<(1-p)(q-1-pq-3p).

From the above discussion, we have the following result.

Theorem 2.2.

Equation (1.1) has a positive prime period-two solution ,ϕ,ψ,ϕ,ψ, if and only if kisodd,p<1,q>1,4r<(1-p)(q-pq-3p-1). Furthermore, if (2.12) holds, then the prime period-two solution of (1.1) is “unique” and the values of ϕ and ψ are the positive roots of the quadratic equation t2-(1-p)t+r+p(1-p)q-1=0.

3. Boundedness and Invariant Intervals

In this section, we discuss the boundedness, invariant intervals of (1.1).

3.1. BoundednessTheorem 3.1.

All positive solutions of (1.1) are bounded.

Proof.

Equation (1.1) can be written as yn+1=r+pyn+yn-kqyn+yn-kpyn+yn-kqyn+yn-kmin{(p/q),1}(qyn+yn-k)qyn+yn-k=min{pq,1} for all n0. We denote K=min{pq,1}. Then yn+1=r+pyn+yn-kqyn+yn-kr+pyn+yn-k(q/2)K+(K/2)+(q/2)yn+(1/2)yn-kmax{r,p,1}(1+yn+yn-k)min{(q/2)K+(K/2),(q/2),(1/2)}(1+yn+yn-k)=max{r,p,1}min{(q/2)K+(K/2),(q/2),(1/2)}   for all n>k. The proof is complete.

Let {yn}n=-k be a positive solution of (1.1). Then the following identities are easily established:

yn+1-1=(q-p)((r/(q-p))-yn)qyn+yn-k,n0,yn+1-pq=((p-q)/q)((qr/(p-q))-yn-k)qyn+yn-k,n0,yn+1-qrp-q=((p2-pq-q2r)/(p-q))(yn+(1/q))qyn+yn-k+((p-q-qr)/(p-q))(yn-k-(p/q))qyn+yn-k,n0,yn+1-p+rq=r(1-yn)+((q-p-r)/q)yn-kqyn+yn-k,n0,yn-yn+2(k+1)=(yn-(p/q))(q2yn+kyn+2k+1+qynyn+2k+1)qyn+2k+1(qyn+k+yn)+(r+pyn+k+yn)+pyn+k(yn-((p+qr)/p))+(yn2-yn-r)qyn+2k+1(qyn+k+yn)+(r+pyn+k+yn),n0.

When q=p+r, the unique positive equilibrium of (1.1) is y̅=1, (3.4) becomes

yn+1-1=r(1-yn)qyn+yn-k,n0.

When p=q(1+1+4r)/2, the unique positive equilibrium is y̅=p/q, (3.5) becomes

yn+1-pq=(p-q)/q((p/q)-yn-k)qyn+yn-k,n0, and (3.8) becomes

yn-yn+2(k+1)=(yn-(p/q))(q2yn+kyn+2k+1+qynyn+2k+1+pyn+k+yn+(p-q)/q)qyn+2k+1(qyn+k+yn)+(r+pyn+k+yn),n0.

Set

f(x,y)=r+px+yqx+y.

Lemma 3.2.

Assume that f(x,y) is defined in (3.12). Then the following statements are true:

assume p<q. Then f(x,y) is strictly decreasing in x and increasing in y for xr/(q-p); and it is strictly decreasing in each of its arguments for x<r/(q-p);

assume p>q. Then f(x,y) is increasing in x and strictly decreasing in y for yqr/(p-q); and it is strictly decreasing in each of its arguments for y<qr/(p-q).

Proof.

By calculating the partial derivatives of the function f(x,y), we have fx(x,y)=(p-q)y-qr(qx+y)2,fy(x,y)=(q-p)x-r(qx+y)2, from which these statements easily follow.

3.2. Invariant Interval

In this subsection, we discuss the invariant interval of (1.1).

3.2.1. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M127"><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></inline-formula>Lemma 3.3.

Assume that p<q, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

yn>p/q for all n1;

If for some N0, yN>r/(q-p), then yN+1<1;

If for some N0, yN=r/(q-p), then yN+1=1;

If for some N0, yN<r/(q-p), then yN+1>1;

If p<q<p+r, then (1.1) possesses an invariant interval [1,r/(q-p)] and y̅(1,r/(q-p));

If p+r<q<p+qr/p, then (1.1) possesses an invariant interval [r/(q-p),1] and y̅(r/(q-p),1);

If qp+qr/p, then (1.1) possesses an invariant interval [p/q,1] and y̅(p/q,1).

Proof.

The proofs of (i)–(iv) are straightforward consequences of the identities (3.5) and (3.4). So we only prove (v)–(vii). By the condition (i) of Lemma 3.2, the function f(x,y) is strictly decreasing in x and increasing in y for xr/(q-p); and it is strictly decreasing in both arguments for x<r/(q-p).

(v) Using the decreasing character of f, we obtain

1=f(rq-p,rq-p)<f(x,y)<f(1,1)=r+p+1q+1<rq-p. The inequalities 1<rq-p,r+p+1q+1<rq-p are equivalent to the inequality q<p+r.

On the other hand, y̅ is the unique positive root of quadratic equation

(q+1)y2-(p+1)y-r=0. Since (q+1)(rq-p)2-(p+1)rq-p-r=r(q+1)(p+r-q)(q-p)2>0,(q+1)-(p+1)-r=q-p-r<0, then we have that y̅(1,r/(q-p)).

(vi) By using the monotonic character of f, we obtain

(q-p)(p+r)+rq2-pq+r=f(1,rq-p)f(x,y)f(rq-p,1)=1. The inequalities (q-p)(p+r)+rq2-pq+r>rq-p,rq-p<1 follow from the inequality q>p+r.

On the other hand, similar to (v) it can be proved that y̅(r/(q-p),1).

(vii) In this case note that r/(q-p)p/q<1 holds, and using the monotonic character of f, we obtain

pq<qr+pq+pq2+p=f(1,pq)f(x,y)f(pq,1)=qr+p2+qq(p+1)1. Furthermore, similar to (v) it follows y̅(p/q,1). The proof is complete.

When q=p+r, (3.9) implies that the following result holds.

Lemma 3.4.

Assume that q=p+r, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

If for some N0, yN>1, then yN+1<1;

If for some N0, yN=1, then yN+1=1;

If for some N0, yN<1, then yN+1>1.

3.2.2. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M184"><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi>q</mml:mi></mml:math></inline-formula>Lemma 3.5.

Assume that p>q, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

yn>1 for all n1;

If for some N0, yN<qr/(p-q), then yN+k+1>p/q;

If for some N0, yN=qr/(p-q), then yN+k+1=p/q;

If for some N0, yN>qr/(p-q), then yN+k+1<p/q;

If q<p<q(1+1+4r)/2, then (1.1) possesses an invariant interval [p/q,qr/(p-q)] and y̅(p/q,qr/(p-q));

If q(1+1+4r)/2<p<q+qr, then (1.1) possesses an invariant interval [qr/(p-q),p/q] and y̅(qr/(p-q),p/q);

If pq+qr, then (1.1) possesses an invariant interval [1,p/q] and y̅(1,p/q).

Proof.

The proofs of (i)–(iv) are direct consequences of the identities (3.4) and (3.5). So we only give the proofs (v)–(vii). By Lemma 3.2 (ii), the function f(x,y) is increasing in x and strictly decreasing in y for yqr/(p-q); and it is strictly decreasing in each of its arguments for y<qr/(p-q).

(v) Using the decreasing character of f, we obtain

pq=f(qrp-q,qrp-q)f(x,y)f(pq,pq)=qr+p(p+1)p(q+1)qrp-q. The inequalities qr+p(p+1)p(q+1)qrp-q,pq<qrp-q are equivalent to the inequality p<q(1+1+4r)/2. That is, [p/q,qr/(p-q)] is an invariant interval of (1.1).

On the other hand, similar to Lemma 3.3 (v), it can be proved that y̅(p/q,qr/(p-q)).

(vi) By using the monotonic character of f, we obtain

qrp-q(qr+p)(p-q)+pq2rq3r+p(p-q)=f(qrp-q,pq)f(x,y)f(pq,qrp-q)=pq. The inequalities (qr+p)(p-q)+pq2rq3r+p(p-q)qrp-q,qrp-qpq are equivalent to the inequality p>q(1+1+4r)/2.

On the other hand, similar to Lemma 3.3 (v) it can be proved that y̅(qr/(p-q),p/q).

(vii) In this case note that qr/(p-q)1<p/q holds. By the monotonic character of f, we have

1<qr+pq+pq2+p=f(1,pq)f(x,y)f(pq,1)=qr+p2+qq(p+1)pq. The inequalities qr+pq+pq2+p>1,qr+p2+qq(p+1)pq are equivalent to the inequality pq+qr.

Furthermore, similar to Lemma 3.3 (v), it follows y̅(1,p/q). The proof is complete.

Lemma 3.6.

Assume that p=q(1+1+4r)/2, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

If for some N0, yN<p/q, then yN+k+1>p/q;

If for some N0, yN=p/q, then yN+k+1=p/q;

If for some N0, yN>p/q, then yN+k+1<p/q;

If for some N0, yN>p/q, then yN>yN+2(k+1)>p/q;

If for some N0, yN<p/q, then yN<yN+2(k+1)<p/q.

Proof.

In this case, we have that qr/(p-q)=p/q. These results follow from the identities (3.10) and (3.11) and the details are omitted.

4. Global Attractivity

In this section, we discuss the global attractivity of the positive equilibrium of (1.1). We show that the unique positive equilibrium y̅ of (1.1) is a global attractor when p=q or p<1 and p<qpq+1+3p or q<p1.

4.1. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M252"><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>q</mml:mi></mml:math></inline-formula>

In this subsection, we discuss the behavior of positive solutions of (1.1) when p=q.

Theorem 4.1.

Assume that p=q holds, and {yn}n=-k is a positive solution of (1.1). Then the unique positive equilibrium y̅ of (1.1) is a global attractor.

Proof.

By the change of variables yn=1+rp+1un, Equation (1.1) reduces to the difference equation un+1=11+(pr/(p+1)2)un+(r/(p+1)2)un-k,n0. The unique positive equilibrium u̅ of (4.2) is u̅=-(p+1)+(p+1)2+4r(p+1)2r. Applying Lemma 1.3 in interval [0,1], then every positive solution of (1.1) converges to u̅. That is, u̅ is a global attractor. So, y̅ is a global attractor.

4.2. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M265"><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></inline-formula>

In this subsection, we present global attractivity of (1.1) when p<q.

The following result is straightforward consequence of the identity (3.7).

Lemma 4.2.

Assume that p<q holds, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

Suppose that q<p+r. If for some N0, yN>1, then yN+1<(p+r)/q;

Suppose that q>p+r. If for some N0, yN<1, then yN+1>(p+r)/q.

Theorem 4.3.

Assume that p<q, p<1 and qpq+1+3p hold. Let {yn}n=-k be a positive solution of (1.1). Then the following statements hold true:

Suppose q<p+r. If y0[1,r/(q-p)], then yn[1,r/(q-p)] for n1. Furthermore, every positive solution of (1.1) lies eventually in the interval [1,r/(q-p)].

Suppose q>p+r. If y0[r/(q-p),1], then yn[r/(q-p),1] for n1. Furthermore, every positive solution of (1.1) lies eventually in the interval [r/(q-p),1].

Proof.

We only give the proof of (i), the proof of (ii) is similar and will be omitted. First, note that in this case p/q<1<(p+r)/q<r/(q-p) holds.

If y0[1,r/(q-p)], then by Lemma 3.3 (iv), we have that y1>1, and by Lemma 4.2 (i), we obtain that y1<(p+r)/q<r/(q-p), which implies that y1[1,r/(q-p)], by induction, we have yn[1,r/(q-p)], for n1.

Now, to complete the proof it remains to show that when y0[1,r/(q-p)], there exists N>0 such that yN[1,r/(q-p)].

If y0[1,r/(q-p)], then we have the following two cases to be considered:

y0>r/(q-p);

y0<1.

Case (a). From Lemma 3.3 (ii), we see that y1<1. Thus, in the sequel, we only consider case (b).

If y0<1, then by Lemma 3.3 (iv), we have y1>1, and from Lemma 4.2 (i), we have y2<(p+r)/q<r/(q-p). So y3>1 and y4<r/(q-p). By induction, there exists exactly one term greater than 1, which is followed by exactly one term less than r/(q-p), which is followed by exactly one term greater than 1, and so on. If for some N>0, 1yNr/(q-p), then the former assertion implies that the result is true.

So assume for the sake of contradiction, that for all n1, yn never enter the interval [1,r/(q-p)], then the sequence {yn}n=1 will oscillate relative to the interval [1,r/(q-p)] with semicycles of length one. Consider the subsequence {y2n}n=1 and {y2n+1}n=1 of solution {yn}n=-k, we have

y2n<1,y2n+1>rq-pforn1. Let L=limnsupy2n,l=limninfy2n,M=limnsupy2n+1,m=limninfy2n+1, which in view of Theorem 3.1 exist as finite numbers, such that Lr+pm+lqm+l,lr+pM+LqM+L,Mr+pl+mql+m,mr+pL+MqL+M. From (4.6), we have q(Lm-lM)p(m-M)+(l-L), which implies that Lm-lM0. Also, from (4.7), we have q(lM-Lm)p(l-L)+(m-M), which implies that lM-Lm0. Thus lM-Lm=0 and L=l, M=m hold, from which it follows that limny2n and limny2n+1 exist.

Set

limny2n=L,limny2n+1=M, then L1, Mr/(q-p) and L, M satisfies the system L=r+pM+LqM+L,M=r+pL+MqL+M, which implies that L, M is a period-two solution of (1.1). Furthermore, in view of Theorem 2.2, (1.1) has no period-two solution when p<1 and qpq+1+3p hold. This is a contradiction, as desired. The proof is complete.

Theorem 4.4.

Assume that p<q, p<1 and qpq+1+3p hold. Then the unique positive equilibrium y̅ of (1.1) is a global attractor.

Proof.

To complete the proof, there are four cases to be considered.

Case (i). q<p+r.

By Theorem 4.3 (i), we know that all solutions of (1.1) lies eventually in the invariant interval [1,r/(q-p)]. Furthermore, the function f(x,y) is non-increasing in each of its arguments in the interval [1,r/(q-p)]. Thus, applying Lemma 1.3, every solution of (1.1) converges to y̅, that is, y̅ is a global attractor.

Case (ii). q=p+r.

In this case, the only positive equilibrium is y̅=1. In view of Lemma 3.4, we see that, after the first semicycle, the nontrivial solution oscillates about y̅ with semicycles of length one. Consider the subsequences {y2n}n=1 and {y2n+1}n=1 of any nontrivial solution {yn}n=-k of (1.1). We have

y2n<1,y2n+1>1,forn1, or vice versa. Here, we may assume, without loss of generality, that y2n<1 and y2n+1>1, for n1.

Let

L=limnsupy2n,l=limninfy2n,M=limnsupy2n+1,m=limninfy2n+1, which, in view of Theorem 3.1, exist. Then as the same argument in Theorem 4.3, we can see that limny2n and limny2n+1 exist.

Set

limny2n=L,limny2n+1=M, then L1, M1. If LM, then, also as the same argument in Theorem 4.3, we can see that L, M is a period-two solution of (1.1), which contradicts Theorem 2.2. Thus L=M, from which it follows that limnyn=1, which implies that y̅=1 is a global attractor.

Case (iii). p+r<q<p+(q/p)r.

By Theorem 4.3 (ii), we know that all solutions of (1.1) lies eventually in the invariant interval [r/(q-p),1]. Furthermore, the function f(x,y) decreases in x and increases in y in the interval [r/(q-p),1]. Thus, applying Lemma 1.3, every solution of converges to y̅, that is, y̅ is a global attractor.

Case (iv). qp+(q/p)r.

In this case, we note that r/(q-p)p/q<1 holds. From Theorem 4.3 (ii) and Lemma 3.3 (i), we know that all solutions of (1.1) eventually enter the invariant interval [p/q,1]. Hence, by using the same argument in (iii), y̅ is a global attractor. The proof is complete.

4.3. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M388"><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi>q</mml:mi></mml:math></inline-formula>

In this subsection, we discuss the global behavior of (1.1) when p>q.

The following three results are the direct consequences of equations (3.4), (3.5), (3.6), and (3.8).

Lemma 4.5.

Assume that q<p<q(1+1+4r)/2, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

If for some N0, yN<p/q, then yN<yN+2(k+1)<qr/(p-q);

If for some N0, yN>qr/(p-q), then p/q<yN+2(k+1)<yN;

If for some N0, p/qyNqr/(p-q), then p/qyN+2(k+1)qr/(p-q).

Lemma 4.6.

Assume that q(1+1+4r)/2<p<q+qr, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

If for some N0, yN<qr/(p-q), then yN<yN+2(k+1)<p/q;

If for some N0, yN>p/q, then qr/(p-q)<yN+2(k+1)<yN;

If for some N0, qr/(p-q)yNp/q, then qr/(p-q)yN+2(k+1)p/q.

Lemma 4.7.

Assume that pq+qr, and {yn}n=-k is a positive solution of (1.1). Then the following statements are true:

yn>1 for n1;

If for some N0, yN>p/q, then 1<yN+2(k+1)<p/q and yN+2(k+1)<yN;

If for some N0, 1<yNp/q, then 1<yN+2(k+1)p/q.

Theorem 4.8.

Assume that p>q holds, and let {yn}n=-k be a positive solution of (1.1). Then the following statements hold true:

If p<q(1+1+4r)/2, then every positive solution of (1.1) lies eventually in the interval [p/q,qr/(p-q)].

If q(1+1+4r)/2<p<q+qr, then every positive solution of (1.1) lies eventually in the interval [qr/(p-q),p/q].

If pq+qr, then every positive solution of (1.1) lies eventually in the interval [1,p/q].

Proof.

We only give the proof of (i), the proofs of (ii) and (iii) are similar and will be omitted.

When q<p<q(1+1+4r)/2, recall that from Lemma 3.5, [p/q,qr/(p-q)] is an invariant interval and so it follows that every solution of (1.1) with k+1 consecutive values in [p/q,qr/(p-q)], lies eventually in this interval. If the solution is not eventually in [p/q,qr/(p-q)], there are three cases to be considered.

Case (i). If for some N0, yN>qr/(p-q), then there are two cases to be considered. If yN+2(k+1)nqr/(p-q) for every nN, then by Lemma 4.5, we have

yN+2(k+1)(n-1)>yN+2(k+1)n>pq, hence, the subsequence {yN+2(k+1)n} is strictly monotonically decreasing convergent and its limit S satisfies Sqr/(p-q). Taking limit on both sides of (3.8), we obtain a contradiction. If for some n0, yN+2(k+1)n0<qr/(p-q), then by Lemma 4.5 we obtain that {yN+2(k+1)n} is eventually in the interval [p/q,qr/(p-q)].

Case (ii). If for some N0, yN<p/q, then there are two cases to be considered. If yN+2(k+1)n<p/q for every nN, then by Lemma 4.5 we obtain

yN+2(k+1)(n-1)<yN+2(k+1)n<qrp-q, which implies that the subsequence {yN+2(k+1)n} is convergent. Then as the same argument in case (i), obtain a contradiction. If for some n0, yN+2(k+1)n0>p/q, then by Lemma 4.5 we have that {yN+2(k+1)n} is eventually in the interval [p/q,qr/(p-q)].

Case (iii). If for some N0,p/qyNqr/(p-q), then by Lemma 4.5 it follows that p/qyN+2(k+1)nqr/(p-q). Assume that there is a subsequence {yN0+2(k+1)n} such that yN0+2(k+1)nqr/(p-q), or yN0+2(k+1)np/q, for every nN. Then its limit S satisfies Sqr/(p-q), or Sp/q. Taking limit on both sides of (3.8), obtain a contradiction. Hence, for all N{1,2,,2(k+1)} the subsequences {yN+2(k+1)n} are eventually in the interval [p/q,qr/(p-q)].

Theorem 4.9.

Assume that p>q and p1 hold. Then the unique positive equilibrium y̅ of (1.1) is a global attractor.

Proof.

The proof will be accomplished by considering the following four cases.

Case (i). p<q(1+1+4r)/2.

By part (i) of Theorem 4.8, we know that all positive solutions of (1.1) lie eventually in the invariant interval [p/q,qr/(p-q)]. Furthermore, the function f(x,y) is nonincreasing in each of its arguments in the interval [p/q,qr/(p-q)]. Thus, applying Lemma 1.3, every solution of (1.1) converges to y̅, that is, y̅ is a global attractor.

Case (ii). p=q(1+1+4r)/2.

In this case, the only positive equilibrium of (1.1) is y̅=p/q. From Lemma 3.6 and (3.11), we know that each of the 2(k+1) subsequences

{y2(k+1)n+i}n=0fori=1,2,,2(k+1) of any solution {yn}n=-k of (1.1) is either identically equal to p/q or strictly monotonically convergent and its limit is greater than zero. Set Li=limny2(k+1)n+ifori=1,2,,2(k+1). Then, clearly, ,L1,L2,,L2(k+1), is a period solution of (1.1) with period 2(k+1). By applying (3.11) to the solution (4.17) and using the fact Li>0 for i=1,2,,2(k+1), we see that Li=pqfori=1,2,,2(k+1), and so limnyn=pq, which implies that y̅=p/q is a global attractor.

Case (iii). q(1+1+4r)/2<p<q+qr.

By Theorem 4.8 (ii), all positive solutions of (1.1) eventually enter the invariant interval [qr/(p-q),p/q]. Furthermore, the function f(x,y) increases in x and decreases in y in the interval [qr/(p-q),p/q]. Thus, applying Lemma 1.3 and assumption p1, every solution of (1.1) converges to y̅. So, y̅ is a global attractor.

Case (iv). pq+qr.

In this case, we note that qr/(p-q)1<p/q holds. In view of Theorem 4.8 (iii), we obtain that all solutions of (1.1) eventually enter the invariant interval [1,p/q]. Furthermore, the function f(x,y) increases in x and decreases in y in the interval [1,p/q]. Then using the same argument in case (iii), every solution of (1.1) converges to y̅. Thus the equilibrium y̅ is a global attractor. The proof is complete.

Finally, we summarize our results and obtain the following theorem, which shows that y̅ is a global attractor in three cases.

Theorem 4.10.

The unique positive equilibrium y̅ of (1.1) is a global attractor, when one of the following three cases holds:

p=q;

p<1 and p<qpq+1+3p;

q<p1.

BasuS.MerinoO.Global behavior of solutions to two classes of second-order rational difference equationsAdvances in Difference Equations200920092710.1155/2009/128602128602MR2538482ZBL1177.39016KulenovićM. R. S.LadasG.Dynamics of Second Order Rational Difference Equations with Open Problem and Conjectures2002Boca Raton, Fla, USAChapman & Hall/CRCxii+218MR1935074KocićV. L.LadasG.Global Behavior of Nonlinear Difference Equations of Higher Order with Applications1993256Dordrecht, The NetherlandsKluwer Academic Publishersxii+228Mathematics and Its ApplicationsMR1247956AgarwalR. P.LiW.-T.PangP. Y. H.Asymptotic behavior of a class of nonlinear delay difference equationsJournal of Difference Equations and Applications20028871972810.1080/1023619021000000735MR1914599ZBL1010.39003AmlehA. M.CamouzisE.LadasG.On the boundedness character of rational equations. IIJournal of Difference Equations and Applications200612663765010.1080/10236190500539279MR2240380ZBL1104.39002CunninghamK.KulenovićM. R. S.LadasG.ValicentiS. V.On the recursive sequence xn+1=(α+βxn)/(Bxn+Cxn1)Nonlinear Analysis: Theory, Methods & Applications20014774603461410.1016/S0362-546X(01)00573-9MR1975854ZBL1042.39522DeVaultR.KosmalaW.LadasG.SchultzS. W.Global behavior of yn+1=(p+ynk)/(qyn+ynk)Nonlinear Analysis: Theory, Methods & Applications20014774743475110.1016/S0362-546X(01)00586-7MR1975867ZBL1042.39523El-AfifiM. M.afifikh@yahoo.comOn the recursive sequence xn+1=(α+βxn+γxn-1)/(Bxn+Cxn-1)Applied Mathematics and Computation200414736176282-s2.0-003542515010.1016/S0096-3003(02)00800-7GibbonsG. H.KulenovićM. R. S.LadasG.LadasG.PintoM.On the dynamics of xn+1=(α+βxn+γxn-1)/(A+Bxn)New Trends in Difference Equaations2002London, UKTaylor & Francis141158HuL.-X.LiW.-T.StevićS.Global asymptotic stability of a second order rational difference equationJournal of Difference Equations and Applications200814877979710.1080/10236190701827945MR2433920ZBL1153.39015HuangY. S.KnopfP. M.Boundedness of positive solutions of second-order rational difference equationsJournal of Difference Equations and Applications2004101193594010.1080/10236190412331285360MR2082679ZBL1064.39006KocićV. L.LadasG.RodriguesI. W.On rational recursive sequencesJournal of Mathematical Analysis and Applications1993173112715710.1006/jmaa.1993.1057MR1205914ZBL0777.39002KulenovićM. R. S.LadasG.MartinsL. F.RodriguesI. W.The dynamics of xn+1=(α+βxn)/(A+Bxn+Cxn-1) facts and conjecturesComputers and Mathematics with Applications2003456–9108710992-s2.0-000021066310.1016/S0898-1221(03)00090-7KulenovićM. R. S.MerinoO.A note on unbounded solutions of a class of second order rational difference equationsJournal of Difference Equations and Applications200612777778110.1080/10236190600734184MR2243836ZBL1107.39007KulenovićM. R. S.MerinoO.Global attractivity of the equilibrium of xn+1=(pxn+xn1)/(qxn+xn1) for q<pJournal of Difference Equations and Applications200612110110810.1080/10236190500410109MR2197588ZBL1099.39007Mazrooei-SebdaniR.DehghanM.Dynamics of a non-linear difference equationApplied Mathematics and Computation20061782250261MR2248485ZBL1106.39011SimsekD.DemirB.CinarC.On the solutions of the system of difference equations xn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}Discrete Dynamics in Nature and Society200920091110.1155/2009/325296325296MR2520418ZBL1178.39013SuY.-H.LiW.-T.StevićS.Dynamics of a higher order nonlinear rational difference equationJournal of Difference Equations and Applications200511213315010.1080/10236190512331319352MR2114321ZBL1071.39017